There is also a free bus between terminals at LAX. Look for the "A" bus on the arrivals level roadway. Some of the other airport shuttle routes that end up in parking lots or train stations (C or G) can also get you form T1 to TBIT, but in the other direction you must wait for an "A".
So here's the quandary with that. If you rent carts at T1 but use the shuttle, you (your family) will have to haul the luggage onto the bus, leave the cart behind when you board the bus, get your luggage OFF the bus when you arrive at TBIT, then do something with your luggage to get to the check-in counter upstairs, meaning you'll have to rent carts AGAIN at TBIT.
(FYI, the A shuttle stops for you are T1 > T2 > T3 > TBIT)
Or, you rent the carts at T1 and make the walk. It'll probably be a nice day along the coast (because it's LA and all), so the walk should take <10 mins. Go up to the upper/departures level for sun and fresh(er) air, or stay on the lower/arrivals level for shade/cover.